Since Space Engine uses procedural generation to fill in unknown celestial bodies, I assume that two players going to the same distant galaxy will find entirely different stars and planets there. So if one sends his recorded locations to the other, it won't show the same planets/stars/terrains, right?

Yes, in the next version you will be able to share locations way more easily.

Unfortunately, as Julian said, mods change the pattern of generation and you might not be able to open the location anymore, that's why I suggest having two versions of SE, a "clean" and a "modded" version. (I did that for many other games, even if with recent games I rather create .bat files to move and restore modded files due to very large size of game data)

The universe is not required to be in perfect harmony with human ambition.

Just curious. How do you feel when someone asks for the location of a planet in one of your Space Engine videos? I hesitate to reveal my planet locations in my You Tube videos. Maybe this sounds strange, but it takes time to search for and find a really beautiful planet. Seems like the person asking for the location should spend his/her own time finding their own special locations. Am I being ridiculous?